Question

The ages of Hari and Harry are in the ratio 5: 7. Four years from now the ratio of their ages will be 3: 4. Find the sum of their present ages.

Answer 1

Hint: To find the sum, we write the present ages in terms of variable x. Then we write the ages after 4 years in terms of variable y. Then we write the relation between the present age and age after 4 years of Hari and Harry in equations of x and y and solve.

Complete step-by-step answer:
Given, the ratio of the present ages of Hari and Harry is 5:7.
Let the present age of Hari is 5x.
Let the present age of Harry is 7x.

Given, the ratio of the ages of Hari and Harry after 4 years is 3:4.
Let the present age of Hari is 3y.
Let the present age of Harry is 4y.

	Hari	Harry
Present age	5x	7x
Age after 4 years	3y	4y

So according to the given, we have

5x + 4 = 3y - (1) and 7x + 4 = 4y - (2)

On solving the two equations we get x and y.

Multiply (1) by 4 and multiply (2) by 3, we get,
${\text{20x + 16 = 12y}}$ And ${\text{21x + 12 = 12y}}$ respectively.

Since the RHS of both the equation is same, we equate the two

⟹20x +16 = 21x +12
⟹x = 4

Now, the present ages of Hari and Harry are 5x and 7x, i.e.

 The present age of Hari is 5x = 5 x 4 = 20 years
 The present age of Harry is 7x = 7 x 4 = 28 years

Hence, the sum of their present ages = 20 + 28 = 48 years.

Note: In order to solve this type of problems the key is to write the given ratio in terms of unknown variables x and y. The main step is writing the relation between the present age and age after 4 years of Hari and Harry in equations. This becomes the clear case of a 2-equation 2-variables problem. Upon solving the two equations we get the values of the variables.